Boundedness and finite-time blow-up in a Keller–Segel chemotaxis-growth system with flux limitation

This paper deals with a parabolic–elliptic Keller–Segel chemotaxis-growth system with flux limitation

$$\begin{aligned} \left\{ \begin{aligned} u_t&=\nabla \cdot ((u+1)^{m-1}\nabla u)- \nabla \cdot (uf(|\nabla v|^{2})\nabla v)+\lambda u-\mu u^k,&\quad x\in \Omega ,t>0,\\ 0&=\Delta v-M(t)+u,&\quad x\in \Omega ,t>0, \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions, where \(\Omega \subset {\mathbb {R}}^N\) is a smoothly bounded domain, \(m\in {\mathbb {R}}\), \(\lambda>0, \mu >0\), \(k>1\), \(M(t):=\frac{1}{|\Omega |} \mathop {\int }\limits _{\Omega } u(x, t) d x\), \(f\left( |\nabla v|^2\right) =(1+|\nabla v|^2)^{-\alpha }, \alpha \in {\mathbb {R}}\). In this framework, it is shown that when \(N\ge 2, m+k>2, k>1, k\ge m\) and

$$\begin{aligned} \alpha >\frac{4N-(m+k)N-2}{4(N-1)}, \end{aligned}$$

then for all nonnegative initial data, the solution is global and bounded in time. Moreover, when \(\Omega \subset {\mathbb {R}}^N\) \((N\ge 5)\) is a ball, if \(1<m<\min \left\{ \frac{2N-4}{N},1-\frac{1}{N}+\frac{1}{N}\sqrt{N^2-4N+1}\right\} \) and the parameters \(\alpha \) and k satisfy suitable conditions, there exist some initial data \(u_{0}\) such that the solution u(x, t) blows up at finite time \(T_{\max }\) in \(L^{\infty }\)-norm sense.